电气工程代写|数字信号过程代写digital signal process代考|ECSE4530

电气工程代写|数字信号过程代写digital signal process代考|Dependencies between Variables within a Random Process

In the field of statistical signal processing, one of the basic questions asked is how the different random variables in a stochastic process are related to one another. To quantify this dependency, we introduce the autocorrelation and autocovariance functions. Auto- and crosscorrelation definitions may vary between text books; the definitions used in this text agree with [7].
The Autocorrelation Function
The autocorrelation function computes correlation of any two random variables that reside in the same random process. The idea is identical to the correlation $r_{x y}$ that we discussed in the section on random variables. We call it a function owing to the presence of the independent indices, $a$ and $b$, which select the random variables in the process $X$.
$$r_x(a, b)=E\left(x_a \cdot x_b^*\right)$$
Again, $a$ and $b$ are indices $-$ from 0 to $7-$ of the two random variables in our random process $X$.

The autocovariance function computes the covariance between any two random variables that reside in the same random process.
$$c_x(a, b)=E\left(\left[x_a-E\left(x_a\right)\right] \cdot\left[x_b-E\left(x_b\right)\right]^*\right)$$
The autocovariance function determines the common variation of two random variables away from their mean, whereas the autocorrelation function determines the common variation of two random variables away from zero. Therefore, if the random variables of our process are zeromean, then the two functions are equivalent.

Clearly, these functions can be applied to either random process $X$ or $Y$. We may also be interested in the common variation between a random variable in process $X$ and one in $Y$ and in that case we rename our statistical function as follows.

电气工程代写|数字信号过程代写digital signal process代考|Stationary Random Processes

Let us once again consider our symbol generator and the accompanying random process $X$, which is composed of a certain number of random variables. Each one of these random variables has a mean, variance and potentially some effect on the other random variables in the process $X$. If these statistics don’t change over time – meaning they are the same now, in five minutes and next month – then the process is said to be stationary. Furthermore, if the linear time-invariant or LTI system we showed earlier also remains unchanged, then the random process $Y$ is also considered to be stationary.

The autocorrelation and autocovariance functions for a stationary random process will no longer depend on the exact time index $a$ and $b$ inside its vector of random variables but on the time difference between them. The variable $x_0$ influences $x_1$ in the same way as variable $x_{100}$ influences $x_{101}$. Thus, for any time offset of $c$ the following expression holds true.
$$r_x(a, b)=r_x(a+c, b+c)$$
Better yet, we express the functions in terms of a difference in time index which we will call $\tau$, where $\tau=a-b$. The location or index, $a$, of the random variable inside our vector is arbitrary.
$r_x(\tau)=r_x(a, a+\tau) \leftarrow$ Autocorrelation Function (stationary RP)
$c_x(\tau)=c_x(a, a+\tau) \leftarrow$ Autocovariance Function (stationary RP)

If the characteristics of the linear system are indeed time-invariant, then the processes $X$ and $Y$ are said to be joint stationary and we may define the following expressions.
$r_{x y}(\tau)=r_{x y}(a, a+\tau) \leftarrow$ Crosscorrelation Function (stationary $R P$ )
$c_{x y}(\tau)=c_{x y}(a, a+\tau) \quad \leftarrow$ Crosscovariance Function (stationary RP)

数字信号过程代考

电气工程代写|数字信号过程代写digital signal process代考|Dependencies between Variables within a Random Process

$$r_x(a, b)=E\left(x_a \cdot x_b^\right)$$ 再次， $a$ 和 $b$ 是指数 $-从 0$ 到 $7-$ 我们随机过程中的两个随机变量 $X$. 自协方差函数计算位于同一随机过程中的任意两个随机变量之间的协方差。 $$c_x(a, b)=E\left(\left[x_a-E\left(x_a\right)\right] \cdot\left[x_b-E\left(x_b\right)\right]^\right)$$

电气工程代写|数字信号过程代写digital signal process代考|Stationary Random Processes

$$r_x(a, b)=r_x(a+c, b+c)$$

$r_x(\tau)=r_x(a, a+\tau) \leftarrow$ 自相关函数 (固定 RP)
$c_x(\tau)=c_x(a, a+\tau) \leftarrow$ 自协方差函数（固定 RP)

$c_{x y}(\tau)=c_{x y}(a, a+\tau) \quad \leftarrow$ 交叉协方差函数（固定 RP)

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